What kind of illumination to use to produce these 3D ball-and-stick atomic structure models? (specular reflection but no shadows)
A few steps that might help Cycles render like this:
Lighting:
I find it's usually best to adjust this first, on neutral-grey surfaces
A single key point-light, with ambient fill provided by a somewhat luminous World. The World has one color to contribute to the lighting, and another to appear in the camera, thus:

The size of the point-light will affect the size and softness of the 'ping' specular dot on the spheres.
Object visibility:
You've got to cripple Cycles on the objects, to prevent shadows being cast, and stop any diffuse bounce of light between them. Switching off everything bar 'Camera' visibility seems to work, in the Object Properties tab &gt; Visibility panel:

Materials:
For convenience, a Principled shader:

.. with 'Roughness' adjusting size and softness of highlight, and maybe a tweak of 'Specular' intensity. A tiny contribution of 'Emission' in the base color, helps the World fill the away-from-light surfaces.
This sort of result: